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The rational QZ method generalizes the QZ method by implicitly supporting rational subspace iteration. In this paper we extend the rational QZ method by introducing shifts and poles of higher multiplicity in the Hessenberg pencil, which is…
In this article we provide a fast computational method in order to calculate the Moore-Penrose inverse of singular square matrices and of rectangular matrices. The proposed method proves to be much faster and has significantly better…
The nonlinear inverse problem of exponential data fitting is separable since the fitting function is a linear combination of parameterized exponential functions, thus allowing to solve for the linear coefficients separately from the…
An efficient algorithm for computing eigenvectors of a matrix of integers by exact computation is proposed. The components of calculated eigenvectors are expressed as polynomials in the eigenvalue to which the eigenvector is associated, as…
We develop the first stochastic incremental method for calculating the Moore-Penrose pseudoinverse of a real matrix. By leveraging three alternative characterizations of pseudoinverse matrices, we design three methods for calculating the…
In this paper, we introduce novel fast matrix inversion algorithms that leverage triangular decomposition and recurrent formalism, incorporating Strassen's fast matrix multiplication. Our research places particular emphasis on triangular…
We address an algorithm for the least squares fitting of a subset of the eigenvalues of an unknown Hermitian matrix lying an an affine subspace, called the Lift and Projection (LP) method, due to Chen and Chu (SIAM Journal on Numerical…
We present a simple yet powerful framework for solving inverse problems by leveraging automatic differentiation. Our method is broadly applicable whenever a smooth cost function can be defined near the true solution, and a numerical…
We develop an accelerated gradient descent algorithm on the Grassmann manifold to compute the subspace spanned by a number of leading eigenvectors of a symmetric positive semi-definite matrix. This has a constant cost per iteration and a…
In this work, we describe, analyze, and implement a pseudospectral quadrature method for a global computer modeling of the incompressible surface Navier-Stokes equations on the rotating unit sphere. Our spectrally accurate numerical error…
Majority of the current dimensionality reduction or retrieval techniques rely on embedding the learned feature representations onto a computable metric space. Once the learned features are mapped, a distance metric aids the bridging of gaps…
We consider the approximate computation of spectral projectors for symmetric banded matrices. While this problem has received considerable attention, especially in the context of linear scaling electronic structure methods, the presence of…
Semidefinite programming is a fundamental problem class in convex optimization, but despite recent advances in solvers, solving large-scale semidefinite programs remains challenging. Generally the matrix functions involved are spectral or…
Based on the spectral divide-and-conquer algorithm by Nakatsukasa and Higham [SIAM J. Sci. Comput., 35(3): A1325-A1349, 2013], we propose a new algorithm for computing all the eigenvalues and eigenvectors of a symmetric banded matrix. For…
A problem that is frequently encountered in a variety of mathematical contexts, is to find the common invariant subspaces of a single, or set of matrices. A new method is proposed that gives a definitive answer to this problem. The key idea…
We propose iterative projection methods for solving square or rectangular consistent linear systems Ax = b. Existing projection methods use sketching matrices (possibly randomized) to generate a sequence of small projected subproblems, but…
Efficient and accurate spectral solvers for nonlocal models in any spatial dimension are presented. The approach we pursue is based on the Fourier multipliers of nonlocal Laplace operators introduced in a previous work. It is demonstrated…
Computing eigenvalues of very large matrices is a critical task in many machine learning applications, including the evaluation of log-determinants, the trace of matrix functions, and other important metrics. As datasets continue to grow in…
We improve the current best running time value to invert sparse matrices over finite fields, lowering it to an expected $O\big(n^{2.2131}\big)$ time for the current values of fast rectangular matrix multiplication. We achieve the same…
This work is to propose an iterative method of choice to compute a stable subspace of a regular matrix pencil. This approach is to define a sequence of matrix pencils via particular left null spaces. We show that this iteration preserves a…